### Abstract

We investigate the long-time behavior of weak solutions to the thin-fllm type equation vt = (xv - vv_{xxx})_{x}, which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form 1/24 (C^{2} - x ^{2})^{2}_{+}, in the norm |||f|||^{2} _{m,1} =∫_{R}(1 + |x|^{2m})|f(x)|^{2} dx + ∫_{R}|f|(x)|^{2} dx. We obtain exponential convergence in the ||| · |||_{m,1} norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H^{1} Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

Original language | English (US) |
---|---|

Pages (from-to) | 4537-4553 |

Number of pages | 17 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 34 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2014 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Keywords

- Euler-Lagrange equation
- Gradient flow
- Lyapunov functional
- Thin-film equation
- Wasserstein distance

### Cite this

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*Discrete and Continuous Dynamical Systems- Series A*, vol. 34, no. 11, pp. 4537-4553. https://doi.org/10.3934/dcds.2014.34.4537

**Localization, smoothness, and convergence to equilibrium for a thin film equation.** / Carlen, Eric A.; Ulusoy, Süleyman.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Localization, smoothness, and convergence to equilibrium for a thin film equation

AU - Carlen, Eric A.

AU - Ulusoy, Süleyman

PY - 2014/11

Y1 - 2014/11

N2 - We investigate the long-time behavior of weak solutions to the thin-fllm type equation vt = (xv - vvxxx)x, which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form 1/24 (C2 - x 2)2+, in the norm |||f|||2 m,1 =∫R(1 + |x|2m)|f(x)|2 dx + ∫R|f|(x)|2 dx. We obtain exponential convergence in the ||| · |||m,1 norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H1 Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

AB - We investigate the long-time behavior of weak solutions to the thin-fllm type equation vt = (xv - vvxxx)x, which arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the form 1/24 (C2 - x 2)2+, in the norm |||f|||2 m,1 =∫R(1 + |x|2m)|f(x)|2 dx + ∫R|f|(x)|2 dx. We obtain exponential convergence in the ||| · |||m,1 norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H1 Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

KW - Euler-Lagrange equation

KW - Gradient flow

KW - Lyapunov functional

KW - Thin-film equation

KW - Wasserstein distance

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UR - http://www.scopus.com/inward/citedby.url?scp=84901749345&partnerID=8YFLogxK

U2 - 10.3934/dcds.2014.34.4537

DO - 10.3934/dcds.2014.34.4537

M3 - Article

AN - SCOPUS:84901749345

VL - 34

SP - 4537

EP - 4553

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 11

ER -